Game One

Game One

Type	Bertrand-Edgeworth
Tournaments	Tournament One

Setup

There are two firms producing a homogeneous good. The demand for the good is given by $q=a-bp$. Without the loss of generality we can set $a=1$ and $b=1$ so that $q=1-p$. Each firm $i$ has the same capacity constraint $k={\frac {2}{3}}$ and in a given round sets price $p_{i}$. We assume efficient rationing. Then if $p_{1}<p_{2}$ we have

$q_{1}=\min(1-p_{1},k)$

$q_{2}=\min(1-q_{1}-p_{2},k)$

If some prices are equal, the demand is split equally among those firms.

In each round the prices are set simultaneously. There are $T=1000$ rounds played in total and each program is allowed to learn from the previous rounds, i.e. all the past prices and earnings are observable.

Suppose there are $P$ players. As long as $P$ is sufficiently small, we simply consider all possible pairs and for each given pair we play $T$ rounds (i.e., a round robin tournament). The profits from all the rounds from all such plays are summed up (no discounting) and the winner is the program with the most profits. The programs are not allowed to learn across different plays, i.e. information learned in playing one competitor cannot be used when playing with another one.

Theory

Profits of firm 1 conditional on prices of firm 2

The first paper that discussed this game when efficient rationing is assumed was Levitan and Shubik (1972). The following exposition is based on their results.

If $k\leq 1/3,$ then there is a pure strategy Nash equilibrium with $p=1-2k$. If $k\geq 1,$ then there is a pure strategy Nash equilibrium with $p=0.$ If $1/3<k<1,$ then no pure strategy equilibria exist, which is illustrated in the Figure for $k=2/3.$ We have that if the price of the second firm $p_{2}>1/2,$ then the first firm responds by setting $p_{1}=1/2,$ and so sells within its capacity constraint. If $1/24<p_{2}\leq 1/2,$ then the first firm responds by undercutting the price just a bit, and sells at capacity. If $p_{2}\leq 1/24,$ then the first firm finds it profitable to charge a monopoly price on the residual demand, namely it sets $p_{1}=1/6.$ Eliminating dominated strategies, we can conclude that the equilibrium prices must be between $1/24$ and $1/2.$ No pure strategy equilibrium exists in this case.

However, there does exist a unique mixed strategy equilibrium. In our case of symmetric capacities, the mixed strategy equilibrium has no atoms and is characterized by the following CDF

$F(p)={\frac {kp-\left({\frac {1-k}{2}}\right)^{2}}{p(p+2k-1)}}$

defined on the support $\left[{\frac {1}{k}}\left({\frac {1-k}{2}}\right)^{2},{\frac {1-k}{2}}\right],$ which translates to $[1/24,1/6]$ for $k=2/3.$

Preventing Tacit Collusion

This game has a unique focal point for tacit collusion, the monopoly price $p={\frac {1}{2}}$. This collusive outcome can be supported with any trigger strategies, e.g. with tit-for-tat. Tacit collusion is not the point of our investigation, therefore we want to prevent it. To this end, the following "trembling hand" rule is imposed. Whenever firms set equal prices, one firm is selected at random and its price is set 1 unit lower than the other price. In theory, this does not change expected payoffs but should prevent tacit collusion. In practice, some collusion might still be observed in the first rounds due to people playing k-level strategies instead of Nash equilibrium strategies, but let us see.

Discussion

This is a copy paste from some email exchanges about the first game, plus recollections of some coffee talks.

Your undercutter does quite well I noticed. I was wondering whether we want to allow that. Giving access to previous rounds opens the door to learning about your opponent's strategy, or to coordinating on collusive outcomes, so we are here studying a repeated game rather than the one-shot game. If repetition is there only to be able to compute expected profits, we would like to exclude that, right? One other issue is this: do we want total profits to determine the ranking? Consider a constant mc simple Bertrand game: Nash strategy gives zero profits, but if you succeed in submitting a monopoly firm and a firm which prices above monopoly, you win the tournament. I am not sure that is what we want to find out. We could also just look at how often you win, for instance. ~ Gijsbert

Regarding your two points, I actually want to play a dynamic game. Couple of reasons, 1) a dynamic game better approximates real industries, think of e.g. airlines, or shipping, 2) in principle, the single period Nash is also an equilibrium in the repeated game, 3) it's just much more interesting to play dynamic games. Indeed, I don't want a collusive outcome, that is boring, so I modified the game just a bit to make it difficult to collude (it's described somewhere on the wiki). I also think the closest literature that we have, on computer tournaments for other games, and on lab experiments for Bertrand-Edgeworth, concerns itself exclusively with repeated games. (I've started discovering some relevant literature and I'll post it on the wiki at some point). In fact, we might want adopt a specification that is directly comparable with one of the lab experiments. ~ Andrei

I was wondering if it would be interesting to also allow the code to (optionally) include a competition of the firms with themselves. e.g. among the simple firms you gave us random performs quit well against another random. Which i find curious for real world implication of "idiotic" collusion. ~ Clemens

What about reputation? More generally speaking, vertical product differentiation. The question is, are we going to observe cyclical behavior? Higher reputation leads to incentives to increase capacity, which might lead to lower quality and then to lower reputation. But reputation touches a lot on consumer behavior. Are consumers fully rational or do they learn quality adaptively? Competition outcomes will depend heavily on these assumptions. Also, fully rational consumers cannot be implemented as NPCs. (How do you program full rationality? :) Or, alternatively, somebody has to play as a representative consumer. ~ Bart